Complex and Nonlocal Systems

该提案涉及复杂和非本地系统，产生于各种背景下，从复杂的流体到社交网络。研究微观物体影响其宏观环境的混合系统是非线性科学的基本组成部分。该项目考虑非线性偏微分方程模型，其中微观部分由动力学方程建模，动力学方程来自基本的随机方程，再加上宏观流体方程。该方案的目的是基本问题的奇点形成，存在性和唯一性或缺乏弱解，混合，长时间的行为和统计解决方案。该提案还涉及主动网络的研究，自然或人工网络的演变，以响应网络依赖的过程。 该提案的总体目标是研究非局部相互作用和反馈对演化复杂系统的动力学影响，解决复杂非局部系统的建模、计算和理论中出现的问题。这样的系统出现在各种各样的情况下，从复杂的物质，如血液或细胞内液，到复杂的关系系统，如社交网络。 复杂物质的特征是存在复杂的、相互作用的网络，这些网络由嵌入、影响并反过来影响基质或溶剂的非常小的物体组成。对这些相互作用的基本理解对于诸如制造具有极端性质的材料或在细胞水平上递送药物等应用的进展至关重要。通过这样的网络的反馈信息的扩散是非常重要的基本生物过程的研究，也为人工网络的研究。该提案的目的是揭示系统规模的质变出现本地和非本地的相互作用的基本机制。

The proposal addresses complex and nonlocal systems, arising in a variety of contexts, ranging from complex fluids to social networks. The study of mixed systems in which microscopic objects influence their macroscopic environment is a fundamental component of nonlinear science. The project considers nonlinear PDE models in which the microscopic part is modelled by kinetic equations derived from underlying stochastic equations, coupled with macroscopic fluid equations. The proposal aims at basic questions of singularity formation, existence and uniqueness or lack thereof for weak solutions, mixing, long time behavior and statistical solutions. The proposal is concerned also with the study of active networks, natural or artificial networks that evolve in response to network-dependent processes. The overarching goal of the proposal is to study the dynamical effects of nonlocal interaction and feedback on evolving complex systems.The proposal addresses problems that arise in the modeling, computation and theory of complex nonlocal systems. Such systems occur in a great variety of circumstances, ranging from complex matter, such as blood or intracellular fluids, to complex relational systems, such as social networks. Complex matter is characterized by the presence of complex, interacting networks of very small objects being embedded, influenced, and in turn influencing a matrix or a solvent. The basic understanding of these interactions is crucial for progress in applications such as the manufacturing of materials with extreme properties or the delivery of drugs at the cellular level. Diffusion of information with feedback through such networks is of great importance for the study of basic biological processes and also for the study of artificial networks. The aim of the proposal is to uncover fundamental mechanisms of system-size qualitative changes emerging from local and nonlocal interactions.